Machine Learning Expectation Maximization and Gaussian Mixtures CSE 473 Chapter 20.3

Feedback in Learning Supervised learning: correct answers for each example Unsupervised learning: correct answers not given Reinforcement learning: occasional rewards

The problem of finding labels for unlabeled data So far we have solved “supervised” classification problems where a teacher told us the label of each example. In nature, items often do not come with labels. How can we learn labels without a teacher? Unlabeled data Labeled data From Shadmehr & Diedrichsen

Example: image segmentation Identify pixels that are white matter, gray matter, or outside of the brain Outside the brain Gray matter White matter Pixel value (normalized) From Shadmehr & Diedrichsen

Raw Proximity Sensor Data Measured distances for expected distance of 300 cm. SonarLaser

Gaussians --   Univariate  Multivariate

© D. Weld and D. Fox 8 Fitting a Gaussian PDF to Data Poor fitGood fit From Russell

© D. Weld and D. Fox 9 Fitting a Gaussian PDF to Data Suppose y = y 1,…,y n,…,y N is a set of N data values Given a Gaussian PDF p with mean  and variance , define: How do we choose  and  to maximise this probability? From Russell

© D. Weld and D. Fox 10 Maximum Likelihood Estimation Define the best fitting Gaussian to be the one such that p(y| ,  ) is maximised. Terminology: p(y| ,  ), thought of as a function of y is the probability (density) of y p(y| ,  ), thought of as a function of ,  is the likelihood of ,  Maximizing p(y| ,  ) with respect to ,  is called Maximum Likelihood (ML) estimation of ,  From Russell

© D. Weld and D. Fox 11 ML estimation of ,  Intuitively: The maximum likelihood estimate of  should be the average value of y 1,…,y N, (the sample mean) The maximum likelihood estimate of  should be the variance of y 1,…,y N. (the sample variance) This turns out to be true: p(y| ,  ) is maximised by setting: From Russell

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If our data is not labeled, we can hypothesize that: 1.There are exactly m classes in the data: 2.Each class y occurs with a specific frequency: 3.Examples of class y are governed by a specific distribution: According to our hypothesis, each example x (i) must have been generated from a specific “mixture” distribution: We might hypothesize that the distributions are Gaussian: Mixtures Parameters of the distributions Mixing proportionsNormal distribution

Hidden variable Measured variable y  Py Graphical Representation of Gaussian Mixtures

Learning of mixture models

Learning Mixtures from Data Consider fixed K = 2 e.g., Unknown parameters  = {  1 ,  1,  2 ,  2,   } Given data D = {x 1,…….x N }, we want to find the parameters  that “best fit” the data

Early Attempts Weldon’s data, n=1000 crabs from Bay of Naples - Ratio of forehead to body length - Suspected existence of 2 separate species

Early Attempts Karl Pearson, 1894: - JRSS paper - proposed a mixture of 2 Gaussians - 5 parameters  = {  1 ,  1,  2 ,  2,   } - parameter estimation -> method of moments - involved solution of 9th order equations! ( see Chapter 10, Stigler (1986), The History of Statistics )

“The solution of an equation of the ninth degree, where almost all powers, to the ninth, of the unknown quantity are existing, is, however, a very laborious task. Mr. Pearson has indeed possessed the energy to perform his heroic task…. But I fear he will have few successors…..” Charlier (1906)

Maximum Likelihood Principle Fisher, 1922 assume a probabilistic model likelihood = p(data | parameters, model) find the parameters that make the data most likely

1977: The EM Algorithm Dempster, Laird, and Rubin General framework for likelihood-based parameter estimation with missing data start with initial guesses of parameters E-step: estimate memberships given params M-step: estimate params given memberships Repeat until convergence Converges to a (local) maximum of likelihood E-step and M-step are often computationally simple Generalizes to maximum a posteriori (with priors)

© D. Weld and D. Fox 24 EM for Mixture of Gaussians E-step: Compute probability that point x j was generated by component i: M-step: Compute new mean, covariance, and component weights:

Anemia Group Control Group

Mixture Density How can we determine the model parameters?

Raw Sensor Data Measured distances for expected distance of 300 cm. SonarLaser

Approximation Results Sonar Laser 300cm400cm

© Daniel S. Weld 37 Hidden Variables But we can’t observe the disease variable Can’t we learn without it?

© Daniel S. Weld 38 We –could- But we’d get a fully-connected network With 708 parameters (vs. 78) Much harder to learn!

© Daniel S. Weld 39 Chicken & Egg Problem If we knew that a training instance (patient) had the disease… It would be easy to learn P(symptom | disease) But we can’t observe disease, so we don’t. If we knew params, e.g. P(symptom | disease) then it’d be easy to estimate if the patient had the disease. But we don’t know these parameters.

© Daniel S. Weld 40 Expectation Maximization (EM) (high-level version) Pretend we do know the parameters Initialize randomly [E step] Compute probability of instance having each possible value of the hidden variable [M step] Treating each instance as fractionally having both values compute the new parameter values Iterate until convergence!